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NEUTRINO PROPERTIES AND DOUBLE $\beta$-DECAY
Ya. B. Zel’dovich, S. Yu. Luk’yanov, Ya. A. Smorodinsky
1. WHAT IS KNOWN ABOUT THE NEUTRINO? SPIN; MASS
One of the most remarkable elementary particles is, undoubtedly, the neutrino.
Its existence was postulated by Pauli on the basis of purely theoretical arguments connected with the requirements of the law of conservation of energy in nuclear $\beta$-decay.
The explanation of the continuous spectrum of $\beta$-particles emitted in the transformation of a nucleus that was in a completely definite state into another nucleus, also in a strictly definite state, presented fundamental difficulties. The question was seriously discussed whether the law of conservation of energy is not violated in $\beta$-decay, and further—whether the law of conservation of energy is applicable at all to elementary processes, and whether it is not merely statistical, valid only on the average, for macroscopic processes. The idea of a new particle—the neutrino—carrying away part of the energy in $\beta$-decay, eliminating these difficulties, was an important stage in the formation of modern physical concepts. At the same time, the properties of the neutrino are such that its direct observation is very difficult: the absence of electric charge and the very small interaction constant with nucleon and electron fields (the magnitude of this constant determines the small probability of the $\beta$-process) make the neutrino an extremely inconvenient object for observation. Nevertheless, our information about the neutrino, though very slowly, is increasing, and at present the prospects for further study of the properties of this particle seem quite real.
What properties of the neutrino can now be regarded as established?
First of all, the neutrino, like the electron and positron, is stable; it does not decay in vacuum; at least, for the existence of such a decay there are no grounds, either theoretical-
cal, or experimental. If this is so, then the neutrino, together with the electron and the proton (one may add: the photon), exhausts the list of stable elementary particles.
Next, the spin of the neutrino is equal to \(1/2\). This follows from the shape of the spectrum of allowed \(\beta\)-decay. If the spin of the neutrino were different from \(1/2\), then the shape of the spectrum would differ from that given by the well-known Fermi formula. For example, if the spin of the neutrino were equal to \(3/2\), then in transitions between nuclei with spins equal to zero, in which the light particles do not carry away the total angular momentum, they would have to carry away orbital angular momentum, since from the spins of the electron \(1/2\) and the neutrino \(3/2\) alone one cannot make zero. It would follow from this that the matrix element would depend on the momenta of the light particles, which, in turn, would lead to a change in the shape of the \(\beta\)-spectrum.
Let us note that from the shape of \(\beta\)-spectra one can also conclude that in \(\beta\)-decay a single neutrino is emitted. Indeed, the emission of several neutral particles would change the statistical weight of the final states in such a way that the mean energy of the electron would be decreased, which would lead to a contradiction with experiment. Direct proof of the emission of one neutrino in \(\beta\)-decay can be obtained by observing the spectrum of recoil nuclei in \(K\)-capture. Since in \(K\)-capture only a neutrino is emitted from the nucleus, the recoil nuclei will all have the same energy if one neutrino is emitted, and will have different energies (depending on the angle between the neutrino directions) if several neutrinos are emitted. Such experiments were carried out by A. I. Alikhanian and A. I. Alikhanov, who first observed in 1941 the recoil nuclei in \(K\)-capture \( \mathrm{Be}^{7} - \mathrm{Li}^{7}\)*). The monochromaticity of the recoil nuclei in this decay was established only in 1952 by Davis. Somewhat earlier, Rodeback and Allen\(^2\) had shown the monochromaticity of recoil nuclei in \(K\)-capture \( \mathrm{A}^{37} \to \mathrm{Cl}^{37}\).
The shape of the \(\beta\)-spectrum also restricts the possible value of the neutrino mass, to which the shape of the spectrum near its upper limit is very sensitive. The most accurately measured shape of the \(\beta\)-spectrum is that for the decay \( \mathrm{H}^{3} \to \mathrm{He}^{3}\) (maximum electron energy \(18\ \mathrm{keV}\)). Analysis of this spectrum gives, for the upper limit of the possible neutrino mass, a value equal to \(1/2000\) of the electron mass\(^3\).
The question of whether the mass of the neutrino is strictly equal to zero, as is true for the mass of the photon, or whether it is simply very small is of great interest. In the first case it would follow from this that the magnetic moment of the neutrino is also equal to zero.
Some considerations on this can be expressed, based on dimensional theory.
*) The first experiments with recoil nuclei in \(\beta\)-decay were carried out by A. Leipunsky in 1936.
The interaction of the neutrino with nucleons and electrons (and also with \(\pi\)- and \(\mu\)-mesons) is the only studied property of the neutrino. For the \(\beta\)-process the energy density of the interaction is given by the formula
\[ H=g\{\psi_1,\ \psi_2,\ \psi_3,\ \psi_4\}, \tag{1} \]
where \(\psi_{1,2,3,4}\) are the wave functions of the four particles; the curly brace symbolically denotes an expression composed of them, with the use of dimensionless operators. The dimension of \(H\) is \(\text{erg}/\text{cm}^3\), the dimension of \(\psi\) is \(\text{cm}^{-3/2}\) (by the normalization condition \(\int \psi^2 d\tau=1\)), whence the dimension of \(g\) is \(\text{erg}\cdot \text{cm}^2\). The value of \(g\) is of order \(2\cdot 10^{-49}\). The interaction (1) leads to the fact that, in the presence of a neutrino, virtual nucleon pairs and electrons are produced in the vacuum
\[ \nu = p + \tilde n + e^- = \nu . \tag{2} \]
Such virtual processes lead to the appearance of a mass and a magnetic moment for the neutrino. The expressions for the mass and magnetic moment in the existing theory are divergent integrals, and substantially worse ones than in electrodynamics. The divergent integrals are obtained by taking into account all virtual processes with all possible values of the momenta of the virtual nucleons. Assuming that the initial expression (1) is valid only for nonrelativistic nucleons (to which the data on \(\beta\)-decay underlying (2) pertain), we cut off the integration at a momentum \(\sim Mc\), where \(M\) is the nucleon mass. The neutrino mass must be proportional to \(g^2\), since vacuum polarization is a second-order process. From dimensions it follows that, with the help of the constants \(M\), \(c\), and \(\hbar\), a mass proportional to \(g^2\) can be formed in only one way:
\[ m_\nu = M(g^2M^4c^2/\hbar^6)\simeq 6\cdot 10^{-10}M\simeq 10^{-6}m_e, \tag{3} \]
\[ m_\nu c^2\simeq 1\ \text{electron-volt}. \]
It may be assumed that the magnetic moment of the neutrino is in the same ratio to the Bohr magneton*). The mass and magnetic moment thus obtained are so small that they do not contradict experiment. However, in the case in which the neutrino and antineutrino coincide, it is evident that the magnetic moment of the neutrino is not merely small, but identically equal to zero, since to every state of polarization of the vacuum (2) there corresponds a charge-conjugate one
\[ \nu = n + \tilde p + e^+ = \nu , \tag{4} \]
*) Nabmias \(^{33}\) showed that the magnetic moment of the neutrino cannot be greater than \(1.8\cdot 10^{-4}\) Bohr magneton. This estimate follows from the absence of ionization due to the magnetic moment of the neutrino (Bethe \(^{34}\)).
giving a contribution to the magnetic moment with the opposite sign. The question arises whether the small value of the mass found is correct, or whether the neutrino mass is identically equal to zero. In the case of light quanta, the theory of the electromagnetic field has such a structure that the mass of the quantum is identically equal to zero, and is not merely small; dimensionally, analogously to (3), we would obtain for the quantum \(m_\gamma=m_e\cdot e^2/\hbar c\), i.e., a rest energy of the order of a kilovolt! However, the so-called gauge invariance of the electromagnetic field leads to the fact that the mass of the quantum is identically equal to zero. Up to the present time the question of the possibility of a theory of a particle with spin \(1/2\) and with mass identically equal to zero (despite interaction with other fields) has not been clarified.
Let us note here the role of the neutrino in cosmic processes. From an estimate of the role of \(\beta\)-processes in the formation of elements it follows that for each proton in the universe there is approximately one neutrino with energy of the order of \(1\) Mev. Near stars, in particular near the Sun, the neutrino flux is considerably greater. Indeed, in both accepted schemes of energy release in stars,
\[ \mathrm{p}+\mathrm{p}=\mathrm{d}+\mathrm{e}^{+}+\nu \qquad \text{and so on.} \tag{5} \]
\[ \mathrm{p}+C_{12}=N_{13}; \qquad N_{13}=C_{13}+\mathrm{e}^{+}+\nu; \qquad \mathrm{p}+C_{13}=N_{14}, \]
\[ \mathrm{p}+N_{14}=O_{15}; \qquad O_{15}=N_{15}+\mathrm{e}^{+}+\nu; \qquad \mathrm{p}+N_{15}=C_{12}+\mathrm{He}_{4} \]
\(\beta\)-processes play an essential role. It is obvious that without \(\beta^{+}\)-processes no cycle of transformation of hydrogen into heavier nuclei, which include neutrons, is possible. The neutrino energy amounts to about 5 or 10% of the energy released in stars. The flux of solar energy onto the surface of the Earth is equal to \(2\ \mathrm{cal}/\mathrm{min}\cdot\mathrm{cm}^{2}\); consequently, the neutrino energy flux is \(\sim 10^{5}\ \mathrm{erg}/\mathrm{sec}\cdot\mathrm{cm}^{2}\); with an average neutrino energy of \(2\cdot10^{-6}\ \mathrm{erg}\), this gives \(5\cdot10^{10}\) particles per second per \(1\ \mathrm{cm}^{2}\). These fluxes are \(10^{7}\)—\(10^{9}\) times greater than the corresponding quantities for cosmic rays. Why, then, does this neutrino flux not manifest itself in any way?
The cross section for interaction of a neutrino with energy of the order of \(1\) Mev with nuclei is of the order of \(10^{-44}\ \mathrm{cm}^{2}\). Therefore the probability of observing a transformation caused by a neutrino is negligible: with a flux of \(5\cdot10^{10}\ \mathrm{cm}^{-2}\mathrm{sec}^{-1}\), in 1 kilogram of matter reaction (5) occurs on the average once per year. The probability that, in passing through the Earth, a neutrino will enter into a reaction is only \(10^{-12}\)*.
The enormous penetrating power of the neutrino leads to interesting fundamental features of thermodynamic equilibrium at high temperature that have been noted in the literature. Let us imagine, for example, a large mass of helium-3 at high tem—
* An estimate of the losses of neutrino energy in the Earth due to the possible magnetic moment of the neutrino is given in the work of Cormack\({}^{35}\).
temperature (of the order of a kilovolt) and high density. In complete thermodynamic equilibrium, in such a mass there must be a definite equilibrium content of tritium, formed when sufficiently fast electrons strike He\(_3\) nuclei and decaying back. However, even a very large system, whose dimensions ensure complete thermal insulation, will be transparent for neutrinos. The reverse process
\[ e^- + \mathrm{He}_3 = T + \nu,\qquad T = \mathrm{He}_3 + e^- + \tilde{\nu} \]
will be accompanied by irreversible losses of energy from the system. The rate of energy leakage does not depend on the dimensions of the system and depends strongly on the temperature, namely it is proportional to the number of electrons which, at the given temperature in the Maxwellian energy distribution, have energies greater than 18 kilovolts (18 kv is the threshold of the reaction \(e^- + \mathrm{He}_3\)).
The example given illustrates the unexpected conclusions to which the unusual properties of the neutrino lead.
2. NEUTRINO AND ANTINEUTRINO
An interesting and important problem, whose solution is being pursued by a number of laboratories in various countries, is the following.
As is known, there exist two types of \(\beta\)-decay: electron decay
\[ \mathrm{n} \to \mathrm{p} + e^- + \tilde{\nu} \tag{1} \]
and positron decay
\[ \mathrm{p} \to \mathrm{n} + e^+ + \nu. \tag{2} \]
Let us note that we are now considering the decay of a nucleon in the field of other nucleons (inside a nucleus), and therefore we need not be concerned with the fulfillment of the laws of conservation of energy and momentum for an individual nucleon\(^*\). Naturally, the question immediately arises whether the neutrinos which fly out in both of these processes and which we have denoted by \(\nu\) and \(\tilde{\nu}\) are one and the same particle, or whether they are two different particles. Neither of these possibilities can be excluded on any general grounds.
From general considerations it follows only that, if the second possibility is realized, the two particles must be related to one another in the same way as the positron and the electron. It is therefore customary to speak of \(\nu\) and \(\tilde{\nu}\) as the neutrino and antineutrino (in the same sense in which the positron is an “antiparticle” with respect to the electron).
\(^*\) For free nucleons, obviously, only neutron decay is possible.
At first glance it might seem that the distinction between particles and antiparticles for neutral particles has a profoundly formal character and is devoid of physical content. However, this is not so.
It is known that the properties of a particle and an antiparticle are related to one another. Namely, it is known that the matrix element for the emission of an antiparticle coincides with the matrix element for the absorption of a particle (with the signs of the energy and momentum reversed). Thus, for example, the creation of a positron with energy \(E\), as is known, is described as the disappearance of an electron with energy \(-E\). Since disappearing particles are written on the left in equations of type (1), and created particles on the right, we can treat equations of types (1) and (2) as algebraic equations, making, when symbols are transferred from one side of the equation to the other, the substitution
\[ e^+ \rightleftarrows e^- \quad \text{and} \quad \nu \rightleftarrows \tilde{\nu}. \]
Using this property, let us compose from reactions (1) and (2) new ones:
\[ e^+ + n \to p + \tilde{\nu}, \tag{3} \]
\[ e^- + p \to n + \nu. \tag{4} \]
These two reactions are nothing other than the reactions of positron and electron capture, of which the latter is observed in the form of capture of an orbital electron. (The second process, although not observed experimentally, cannot yield anything new.) Processes (3) and (4) are not of interest to us, since they merely determine the connection between the matrix elements of \(\beta\)-decay and electron capture.
Let us consider the following two processes, inverse to the preceding ones:
\[ \nu + n \to p + e^-, \tag{5} \]
\[ \tilde{\nu} + p \to n + e^+. \tag{6} \]
These are processes of \(\beta\)-decay induced by neutrinos, i.e. occurring forcibly in a neutrino flux. It is immediately clear that induced \(\beta^-\)-decay can be caused only by a flux of particles emitted in \(\beta^+\)-decay (for definiteness, we shall call precisely these particles neutrinos), while induced \(\beta^+\)-decay can be caused only by a flux of antineutrinos (particles emitted in \(\beta^-\)-decay)*.
At this point we encounter an essential physical difference between the two views of the nature of the neutrino. Indeed, if \(\nu\) and \(\tilde{\nu}\) are different particles, then one may assert that it is impos-
* The question of which of the particles is to be defined as the neutrino and which as the antineutrino is immaterial, since replacing a particle by an antiparticle does not change the formulas, provided only that the mass of the particles is equal to zero.
the following two processes are possible:
\[ \tilde{\nu}+n \to p+e^-, \tag{7} \]
\[ \nu+p \to n+e^+, \tag{8} \]
i.e., for example, that the neutrino flux from a nuclear reactor (originating from chains of \(\beta\)-decays of fragments) cannot cause electron decay.
If, however, the neutrino and antineutrino are one and the same particle, then processes (5) and (6), as well as processes (7) and (8), are equally possible. Thus there arises the possibility of experimentally clarifying this property of the neutrino.
However, a direct experiment to detect, for example, reaction (7) is very difficult. It is true that in 1953 there appeared a preliminary report by Reines and Cowan on attempts to observe reaction (6)—the induced decay of a proton under the influence of a beam of neutrinos from a reactor.^4 In the experiment the process recorded was
\[ \tilde{\nu}+p=n+e^+. \tag{9} \]
For the counting, coincidences were used between two gamma quanta produced in positron annihilation and a gamma quantum arising upon capture of a neutron by cadmium introduced into the system. It is obvious that process (9) is the direct inverse of process (3), which, in turn, is merely another notation for the ordinary \(\beta\)-process (1). Therefore the existence and cross section (of order \(10^{-44}\ \mathrm{cm}^2\)) of the process (9) observed in the experiment could have been predicted theoretically with complete certainty. While confirming the very fact of the existence of the neutrino, the experiments performed did not provide new information about this particle.
To decide the question of the existence of two different particles (the neutrino and the antineutrino) or of their identity, it is necessary to investigate the process
\[ \tilde{\nu}+n=p+e^-, \]
caused by a flux of \(\tilde{\nu}\) produced in a reactor. In this case one need not have in mind interaction necessarily with free neutrons—it would be sufficient to investigate a process with bound neutrons, for example \(\tilde{\nu}+d=2p+e^-\), or an analogous process with the formation of a radioactive element that could be isolated. Such investigations have not yet been carried out.
We can illustrate the difference between the two possibilities (\(\nu=\tilde{\nu}\) and \(\nu\ne\tilde{\nu}\)) by another well-known example. The existence has now been established of three different types of \(\pi\)-mesons: \(\pi^+\), \(\pi^0\), \(\pi^-\), which can transform into one another under
in scattering on nucleons:
\[ \pi^+ + n \rightleftarrows \pi^0 + p;\qquad \pi^0 + n \rightleftarrows \pi^- + p. \]
In this sense one says that the \(\pi^+\), \(\pi^0\), and \(\pi^-\) mesons are three different charge states of one particle*).
If \(\nu = \tilde{\nu}\), then, in an analogous way, both reactions are possible
\[ e^+ + n \rightleftarrows \nu + p,\qquad \nu + n \rightleftarrows p + e^-, \]
i.e., in the same sense one may speak of the triad \(e^-\), \(\nu\), \(e^+\).
On the contrary, if \(\nu \ne \tilde{\nu}\), then, in accordance with what was said above, the reactions
\[ e^+ + n \rightleftarrows \tilde{\nu} + p \]
and
\[ e^- + p \rightleftarrows \nu + n, \]
are possible, i.e., in this case only the transformations
\[ e^+ \rightleftarrows \tilde{\nu}\quad \text{and}\quad e^- \rightleftarrows \nu \]
are possible.
In this case one may speak only of two “charge-conjugate” pairs
\[ (e^+, \tilde{\nu})\quad \text{and}\quad (e^-, \nu). \]
Neutral particles with spin \(1/2\) and with small or zero mass are also produced in the decay of charged \(\pi\)-mesons and \(\mu\)-mesons and in the capture of \(\mu\)-mesons by nuclei:
\[ \pi^- = \mu^- + \nu, \tag{10} \]
\[ \mu^- = e^- + 2\tilde{\nu}, \tag{11} \]
\[ \mu^- + p = n + \tilde{\nu}. \tag{12} \]
It is customary to assume that these particles are identical with the neutrino produced in the \(\beta\)-process, since there are no grounds for the opposite view. Experiment makes it possible to establish the upper limit of the mass of the neutral particles produced in processes (10), (11), (12) only with an accuracy of several electron masses. An indirect confirmation of the identity of the neutrino of the \(\beta\)-process and of processes (10), (11), (12) is the equality of the probabilities of all 4 processes, referred to a unit volume of phase space (per one final state), i.e., the approximate equality of the interaction constants for these processes.
*) It is required here that the interaction of all three types of \(\pi\)-mesons with nucleons be described by the same constant (“hypothesis of charge invariance”).
In a theory in which the neutrino and the antineutrino are distinguished, the question arises as to which—particle or antiparticle—appears in each of the processes (10), (11), (12). The choice made above is justified in a paper by one of the authors[^5].
3. DOUBLE BETA DECAY
In 1939 Fermi[^6] pointed out the possibility of clarifying the neutrino–antineutrino relationship by studying double β-decay. However, Fermi incorrectly estimated the probability of this process. The correct estimate of this probability was given by Sliv[^7]. It is known that in nature there exists a comparatively large number of stable isobaric nuclei with the same mass number and with charges differing by two units, such as, for example, \(\mathrm{Ni}^{64}—\mathrm{Zn}^{64}\). The existence of such isobars is, generally speaking, connected with the fact that the intermediate isobar (in our example \(\mathrm{Cu}^{64}\)) has a mass greater than both end members*), and therefore the transformation of one of the isobars into the other, lighter one, cannot proceed by means of two successive β-decays. It must be borne in mind, however, that not in all cases is the existence of isobars of this kind connected with a large mass of the intermediate nucleus, as is often asserted. In fact, the mass of the intermediate nucleus has not always been measured experimentally and confirms such an explanation. Moreover, there are several cases (\(\mathrm{Ca}^{48}—\mathrm{Ti}^{48}\), for example) in which the mass of the intermediate nucleus turns out to lie between the masses of the two nuclei, and their apparent stability is simply connected with a high degree of forbiddenness (with a long lifetime)**).
Data on this will be given below in Table III.
It is evident that when the transition to a neighboring nucleus is energetically impossible or else very strongly forbidden, the transformation can occur only with the simultaneous emission of two electrons (or positrons). It is not difficult to see that the probability of such a process depends essentially on the assumption concerning the properties of the neutrino. Indeed, if \(\nu \ne \bar{\nu}\), then the antineutrino emitted (in the intermediate state) upon emission of the first of the two electrons cannot be absorbed again by another neutron in the nucleus
*) This property, as is well known, is connected with the fact that nuclei with even numbers of protons and neutrons (Ni and Zn in our example) possess enhanced stability relative to neighboring nuclei.
**) In the case of \(\mathrm{Ca}^{48}\), the transition to \(\mathrm{Sc}^{48}\) is associated with a very small energy (0.3 MeV) and a large change in spin (\(\Delta s \ge 5\)). Therefore the lifetime for such a transition is substantially greater than the assumed lifetime for neutrinoless double β-decay, and this transition has not been observed experimentally. Only when geological data are considered (see below) can single decays with a high degree of forbiddenness influence the interpretation of experiments.
and the emission of the second electron must be accompanied by the emission of one more antineutrino. In this case the reaction proceeds according to the scheme
\[ A \to B + 2e + 2\tilde{\nu}. \tag{1} \]
This decay scheme is called the Fermi scheme*).
The reaction, however, can proceed by a much more probable path if the neutrino has no antiparticle. In this case the neutrino emitted in the decay of the first neutron may be absorbed in the decay of the second neutron (in the same nucleus), and the decay will proceed according to the scheme:
\[ A \to B + 2e^{-} \tag{2} \]
(with no neutrino emission at all).
That this second process is considerably more probable than the first is not immediately obvious, since both processes occur in the same order (namely, in the second there are two radiations) in the constant \(g^{2}\), and the difference is connected only with the magnitude of the matrix elements. Qualitatively this can be seen from the following considerations. In process (1) the number of states corresponding to the emission of two electrons is determined by the volume of the phase space of two neutrinos, bounded by the total value of the decay energy.
In the decay according to the second scheme the neutrino appears only in an intermediate state; its energy may even be greater than the total energy of double \(\beta\)-decay. In this scheme the neutrino energy is limited only by the condition that the neutrino wave function not be alternating in sign inside the nucleus—otherwise the contribution of the corresponding intermediate state to the matrix element and the transition probability sharply decrease.
This condition
\[ \bar{\lambda} > R \tag{3} \]
leads to the fact that the energy of the neutrinos giving the main contribution turns out to be bounded by the value
\[ E < \frac{\hbar c}{R}, \tag{4} \]
\[ E < \hbar c/1.2 \cdot 10^{-13} A^{1/3} = 160/A^{1/3}\ \text{Mev}. \tag{5} \]
*) Sometimes in the literature the Fermi scheme and the Dirac scheme are distinguished. In the first, the emission of an electron is accompanied by the emission of a neutrino (in the Hamiltonian the product of wave functions \(\varphi_e^{+}\varphi_\nu^{+}\)). In the second, the emission of an electron is accompanied by the emission of an antineutrino (in the Hamiltonian the product \(\varphi_e^{+}\varphi_\nu\)). As we have already indicated, if the neutrino mass is equal to zero, then both schemes give identical results.
For medium nuclei this amounts to a quantity of the order of 40 MeV. Thus, in the second scheme the neutrino energy turns out to be many times greater than its energy in the first case. This circumstance leads to an increase in the phase volume of the intermediate state and, correspondingly, to an increase in the probability of decay.
4. PARITY OF ELECTRONS AND NEUTRINOS
The question of the identity or difference of the neutrino and the antineutrino is connected with yet another property of these particles—namely, with their parity. It is known that the wave function of a scalar particle, i.e. a particle without spin, under reflection in the origin of coordinates may either remain unchanged or change sign to the opposite one*):
\[ I_s\psi=+\psi, \]
or
\[ I_s\psi=-\psi. \tag{1} \]
In the first case one says that the particle is even ($\psi$ is a true scalar), and in the second, that it is odd ($\psi$ is a pseudoscalar). An example of an odd particle is the $\pi$-meson, and an example of an even one is the $\alpha$-particle.
In the case of particles described by spinors (spin $1/2$), the situation becomes more complicated. In this case the usual condition $I_s^2=1$ (a reflection carried out twice is the identity operation) must be replaced by the condition
\[ I_s^2=\pm 1, \]
for it is well known that spinors are defined only up to a sign**).
Further, since the wave function consists of several components, the operator $I$ is, generally speaking, a certain matrix. Namely, it can be shown (see the Appendix) that, up to a factor, $I$ is simply the Pauli matrix $\gamma_4$. Thus:
\[ I_s=a\gamma_4, \tag{2} \]
where $a$ is a number satisfying the condition $a^2=\pm 1$, i.e. equal to $1$, $-1$, $i$, $-i$. At first glance it might seem that for spinors, in contrast to scalar functions, there can be four different “parities.” However, it is easy to see that this is not so***).
*) The question of the behavior of wave functions of particles under various transformations is considered in detail by I. S. Shapiro^30.
**) A rotation of the coordinate system through $360^\circ$ changes the sign of a spinor.
***) This circumstance was pointed out by L. D. Landau. See also the article by Kaniellio^32. The latter work, however, contains an erroneous conclusion about the formal impossibility of the Majorana scheme.
Indeed, the product of any two spinors describing physical particles must be either a scalar or a pseudoscalar, i.e., under reflection at the origin of coordinates it must be multiplied either by \(+1\) or by \(-1\). It follows at once that for all particles with spin \(1/2\) there may belong either the operator
\[ I_s=\pm\gamma_4 \quad \text{(i.e. } I^2=1), \tag{3} \]
or the operator
\[ I_s=\pm i\gamma_4 \quad \text{(i.e. } I^2=-1). \tag{4} \]
It turns out that the Majorana scheme \((\nu\equiv\widetilde{\nu})\) can exist only in the case in which (4), and not (3), holds. This is shown as follows. From the properties of the transformations of the Dirac equation set forth in the Appendix it follows that, if some particle with spin \(1/2\) has parity \(a\) (in the sense of the operator (2)), then its antiparticle must have parity \(-a^*\) (the complex-conjugate quantity taken with the opposite sign). It is clear that a particle can coincide with its antiparticle only when \(a\) and \(-a^*\) coincide with one another, which is possible only for imaginary \(a^*\). Thus the proof that \(\nu\equiv\widetilde{\nu}\) would mean that for spinors the transformation (4) takes place.
Below in this article all experimental data obtained up to the present time on double \(\beta\)-decay will be considered in detail.
All published investigations have not led to an unambiguous answer to the question posed. Only the reports, appearing in 1954, on the experiments of MacCarthy \(^{23}\), who observed double \(\beta\)-decay of \(\mathrm{Ca}^{48}\), apparently make it possible, finally, to choose between the two schemes in favor of the scheme \(\nu\equiv\widetilde{\nu}\). A final assessment of the situation can be made only after this work appears in print.
The considerations on parity set forth make it possible to establish the state of the electrons emitted in \(2\beta\)-decay.
Let us consider a decay in which both nuclei, the initial and the final, are even and have spin zero. Only such a case can be realized experimentally **). Then the two electrons must form an even system with angular momentum equal to zero.
*) Let us note that the parity of the particle–antiparticle system does not depend on the arbitrary choice of transformation (3) or (4). In both cases the intrinsic parity is equal to \(-1\), so that, for example, positronium \((e^+ + e^-)\) in the \({}^{1}S_0\)-state is a pseudoscalar particle in any representation.
**) Decay to an excited state (whose spin is usually equal to two) corresponds to a considerably smaller energy (and in most cases is simply impossible energetically) and, correspondingly, to a considerably longer lifetime.
Since the intrinsic parity of such a system is equal to \(i^2=-1\), the electrons must be in the states \(S_{1/2}\) and \(P_{1/2}\). In this case it turns out that the distribution over the angle between the wave vectors will not be isotropic. The form of this correlation was calculated by Primakoff\(^8\). For the adopted variant of the theory of \(\beta\)-decay the correlation is determined by the factor \(1+\cos\vartheta\).
5. PROBABILITY OF DOUBLE \(\beta\)-DECAY
In this paragraph we shall derive a formula for the probability of double \(\beta\)-decay in the scheme \(\nu \equiv \bar{\nu}\). The derivation of the formula differs from that available in the literature\(^ {6,7}\) and leads to a result more convenient for analysis.
Let us first introduce the wave function of the neutrino. Using the definition of the charge-conjugation operator \(C\), i.e. the operator transforming the wave function of a particle into the wave function of the antiparticle (see the Appendix), we construct the function
\[ \psi_\nu = 2^{-1/2}\left(\varphi_\nu + C\bar{\varphi}_\nu\right). \tag{1} \]
Here \(\varphi_\nu\) is the solution of the Dirac equation for a particle with spin \(1/2\) and mass equal to zero, while \(C\bar{\varphi}_\nu\) is the solution of the charge-conjugate equation. Obviously,
\[ C\bar{\psi}_\nu = \psi_\nu, \tag{2} \]
i.e. the wave functions of the neutrino and antineutrino are identical.
The probability of decay is determined by the usual formula of perturbation theory (in the system of units in which \(\hbar=c=1\))
\[ w = 2\pi |V|^2 \rho_F . \tag{3} \]
The matrix element \(V\) is the second-order matrix element of perturbation theory (emission of two electrons) for the operator of \(\beta\)-interaction, and \(\rho_F\) is the statistical weight of the final state.
As is known, there are several possible variants of the theory of \(\beta\)-decay, corresponding to different possibilities for constructing the interaction operator.
In particular, scalar (\(S\)), vector (\(V\)), tensor (\(T\)), pseudovector (\(A\)), and pseudoscalar (\(P\)) variants of the theory are distinguished. Analysis of the shape of \(\beta\)-spectra leads to the conclusion that, at least, the principal forms of the interaction in \(\beta\)-decay are the interactions \(S\) and \(T\), the weights of both types of interaction being approximately the same (see, for example,\(^{9,39}\)*).
* This does not exclude the possibility that the interaction \(V\), rather than \(S\), is realized. This, however, does not change the results for the probability of double \(\beta\)-decay.
Taking this into account, we shall consider only these two interactions. Since, moreover, they do not interfere with one another (see below), we can consider them separately.
Let us denote the interaction operator for the heavy and light particles, respectively, by \(O^{h}\) and \(O\) (for the scalar variant \(O^{h}=1\), for the tensor one \(O=\frac{1}{2}\gamma_l\gamma_k\), the latter operator being normalized so that, in passing to the nonrelativistic case, no superfluous numerical factors appear from terms of the type \(\gamma_1\gamma_2\) and \(\gamma_2\gamma_1\)). The Hamiltonian of the problem has the form:
\[ V=\frac{g_0}{\sqrt{2}}\sum (\bar{\Psi},\,O_i^h\tau_i^{+}\Psi)(\bar{\psi}_e,\,O\psi_\nu), \tag{4} \]
where \(O_i^h\) is an operator acting on the \(i\)-th neutron, \(\tau_i^{+}\) is an operator transforming the \(i\)-th neutron into a proton, \(\Psi\) is the wave function of the nucleus, \(\psi_e\) and \(\psi_\nu\) are the wave functions of the electron and neutrino, respectively, and, finally, \(g_0\) is a constant, whose magnitude we shall discuss below.
Substituting into (4) expression (2) for the neutrino wave function, we obtain:
\[ V=\frac{g_0}{\sqrt{2}}\sum_i\left[(\bar{\Psi}_i,\,O_i^h\tau_i^{+}\Psi_i)(\bar{\psi}_e,\,O\varphi_\nu)- \right. \]
\[ \left. -(\bar{\Psi}_i,\,O_i^h\tau_i^{+}\Psi_i)(\bar{\varphi}_\nu,\,OC,\,\bar{\psi}_e)\right]. \tag{5} \]
In the derivation we used, for transforming the second factor in the second term, the following chain of transformations:
\[ (\bar{\psi}_e,\,OC\bar{\varphi}_\nu)= (\bar{\varphi}_\nu,\,C^{T}O^{T}\bar{\psi}_e)= -(\bar{\varphi}_\nu,\,OC\bar{\psi}_e), \tag{6} \]
the last of which is carried out by virtue of the properties of the matrix \(C\) indicated in the appendix,
\[ C^{T}=-C \quad \text{and} \quad CO^{T}=OC \tag{7} \]
for the scalar and tensor variants.
It is convenient to make such a transformation in order that, in the matrix elements, the operator of neutrino emission \(\bar{\varphi}_\nu\) and the operator of neutrino absorption \(\varphi_\nu\) stand on different sides. In this case the summation over the spins of the virtual neutrino is carried out by a simple passage to matrix multiplication.
From the Hamiltonian thus written it is seen that we must associate with the first act—the emission of an electron accompanied by the emission of a neutrino—the operator
\[ -\frac{g_0}{\sqrt{2}}\sum O_i^h\tau_i^{+}OC, \tag{8} \]
and to the second act—the emission of an electron, accompanied by absorption of a neutrino—the operator
\[ \frac{g_0}{\sqrt{2}}\sum_i O_i^n \tau_i^+ O . \tag{9} \]
According to the usual diagram technique used in quantum electrodynamics, the virtual neutrino corresponds to the operator:
\[ \hat{k}^{-1}=\frac{i\gamma\mathbf{k}-\gamma_4 k_0}{\mathbf{k}^2-k_0^2}, \tag{10} \]
where \(\mathbf{k}\) and \(k_0\) are the momentum and energy of the neutrino (in the intermediate state \(\mathbf{k}^2\ne k_0^2\)).
Let us now construct, according to the usual rules, the matrix element for the emission of an electron and a neutrino with momentum \(\mathbf{k}\) by the \(i\)-th neutron and the emission of a second electron and absorption of the neutrino by the \(k\)-th neutron:
\[ V_{ik}=-\frac{1}{2}g_0^2 \langle F|O_i^n\cdot O_k^n\tau_i^+\tau_k^+ e^{i\mathbf{k}(\mathbf{r}_i-\mathbf{r}_k)}|I\rangle \times \]
\[ \times \langle 2|\hat{O}\hat{k}^{-1}OC|0\rangle . \tag{11} \]
The first factor is the matrix element for the transition of the nucleus from the initial state \(I\) to the final state \(F\). The factor \(e^{i\mathbf{k}(\mathbf{r}_i-\mathbf{r}_k)}\) appears because the neutrino is emitted at the point \(\mathbf{r}_i\), where the \(i\)-th neutron is located, and is absorbed at the point \(\mathbf{r}_k\), where the \(k\)-th neutron is located. We neglect the momenta of the electrons. The expression \(\langle 2|\ldots|0\rangle\) denotes the matrix element of the operator standing in the middle for the transition between the initial state, in which there are no electrons, and the final state, in which there are two electrons. In the usual notation this expression is equal to
\[ \bar{\psi}(2)\hat{O}\hat{k}^{-1}OC\bar{\psi}(1). \]
We must first of all take into account the Pauli principle. For this it is necessary to subtract from expression (11) exactly the same expression in which the order of emission of the electrons is interchanged. (In this case, obviously, the probability of emission of two electrons with identical quantum numbers will be equal to zero.)
In order to simplify the calculation, we shall replace the fourth component of the neutrino wave vector \((k_0)\) by its values determined by the \(\delta\)-function of energy\(*\). Put
\[ k_0=E_i-E'-E_1, \tag{12} \]
\[ \text{*) As is known, in the diagram method the law of energy conservation is carried out formally for the intermediate state, but } \mathbf{k}^2\ne k_0^2,\ \text{i.e., the rest mass of the intermediate particle is as if not conserved. In the old formalism the mass of the particle is unchanged, while the energy is not conserved.} \]
where \(E_i\) and \(E'\) are the energies of the nucleus in the initial and intermediate states, and \(E_1\) is the energy of the electron. Since all these quantities are of the order of several MeV\(^*\), the quantity \(k_0^2\) may be neglected in the denominator of (10) in comparison with \(k^2\) (\(k\) corresponds to an energy of about \(40\) MeV, as indicated above).
The operator (10) for the process in which the second electron is produced first will then differ only in that, in the numerator, instead of the value (12), one must substitute the value
\[ k_0 = E_I - E' - E_2, \tag{13} \]
where \(E_2\) is the energy of the second electron.
Taking the difference of the two operators (it is easy to see that only they will be different in the matrix elements for the two processes), we obtain for the effective operator \(\hat{k}^{-1}\) the expression
\[ \hat{k}^{-1} = (E_1 - E_2)\,\mathbf{k}^{-2}\gamma_4 . \tag{14} \]
Now the matrix element \(V_{ik}\) (11) assumes the form:
\[ V_{ik}(\mathbf{k}) = -\frac{1}{2} g_0^2 (E_1-E_2) \times \left\langle F \left| O_i^{h} O_k^{h}\tau_i^+ \tau_k^+ \frac{e^{i\mathbf{k}(\mathbf{r}_i-\mathbf{r}_k)}}{\mathbf{k}^2} \right| I \right\rangle \times \langle 2|O\gamma_4 OC|0\rangle . \tag{15} \]
For further simplification let us note that the dependence of this matrix element on the neutrino momentum is analogous to the dependence of the Fourier component of the Coulomb interaction. As is known, the potential \(U=1/r\) satisfies the equation \(\Delta U = 4\pi\delta(\mathbf{r})\) and has Fourier components of magnitude
\[ U_{\mathbf{q}}=\frac{4\pi}{\mathbf{q}^{2}} . \tag{16} \]
Using this circumstance, we can immediately integrate (15) over \(k^2\) (perform the inverse Fourier transform). The result of such integration (with multiplication by the usual factor \((2\pi)^{-3}\)) evidently gives the factor
\[ \frac{1}{4\pi}\,|\mathbf{r}_i-\mathbf{r}_k|^{-1}. \tag{17} \]
In the further calculation we shall replace the distance \(|\mathbf{r}_i-\mathbf{r}_k|\) by some mean radius \(R\), which we shall estimate (somewhat
\(^*\) The intermediate state of the nucleus is the state in which one of the neutrons, having turned into a proton, has occupied its final state, while the neutron that is next to undergo the transition will make the other transition. The energy of such a nuclear state corresponds to an excitation of several MeV.
thereby decreasing the probability) as the nuclear radius. Then we obtain:
\[ V_{ik}=-\frac{1}{8\pi}g_0^2\frac{1}{R}(E_1-E_2) \left\langle F\left|O_i^h O_k^h \tau_i^+ \tau_k^+\right|I\right\rangle \langle 2|O\gamma_4OC|0\rangle . \tag{18} \]
Let us now disclose the meaning of the operators \(O^h\) and \(O\). For the scalar variant of the theory the result is obvious:
\[ V_{ik}^s=-\frac{1}{8\pi}\frac{1}{R}(E_1-E_2)g_s^2 \left\langle F\left|\tau_i^+\tau_k^+\right|I\right\rangle \langle 2|\gamma_4 C|0\rangle . \tag{19} \]
For the calculation of the tensor variant we must take into account that the transition takes place between nuclei with spins equal to zero*). Therefore the product \(O_i^h O_k^h\) must form a scalar operator. Consequently, in the product of two tensor operators only the scalar product \(\sigma_i\sigma_k\) plays an essential role, while terms of the type \(\sigma_{lx}\sigma_{ky}\ldots\) may be discarded. Noting further that in transforming the electronic matrix element we can make use of the fact that \(\sigma\) and \(\gamma_4\) anticommute with one another and that \(\sigma_x^2=\sigma_y^2=\sigma_z^2=1\), we obtain for \(V_{ik}\) in the tensor variant:
\[ V_{ik}^T=+\frac{1}{8\pi}\frac{1}{R}(E_1-E_2)g_T^2 \left\langle F\left|\sigma_i\sigma_k\tau_i^+\tau_k^+\right|I\right\rangle \langle 2|\gamma_4 C|0\rangle . \tag{20} \]
As we have already said, the decay probability is determined by a linear combination of both expressions (19) and (20)**). Since in the expression for ordinary \(\beta\)-decay both terms enter with the same weight, it follows that
\[ g_s^2=g_T^2=\frac{1}{2}g^2 . \]
Adding both expressions and summing over all neutrons of the nucleus, we obtain:
\[ V=\frac{1}{16\pi}\frac{1}{R}(E_1-E_2)g^2 \left\langle F\left| \sum_{i\ne k}(1+\sigma_i\sigma_k)\tau_i^+\tau_k^+ \right|I\right\rangle \times \]
\[ \times \langle 2|\gamma_4 C|0\rangle . \tag{21} \]
The decay probability is now computed from formula (3). The calculation of \(\langle 2|\gamma_4 C|0\rangle\) is somewhat cumbersome, and we shall present here
*) Consideration of a transition with a change of spin, i.e. to an excited state of the daughter nucleus, is of no practical significance, since such a transition (if it is at all possible) corresponds to a small energy. Moreover, since the first excited states of even-even nuclei have spin 2, such a transition is possible only in the next order of forbiddenness.
**) It is now clear why the two variants do not interfere with one another. This is a consequence of the requirement of scalar character of the operator; the interference terms would obviously be linear in \(\sigma\).
only the finite result. Namely, substituting the value of the wave functions of the free electrons and integrating over the angle between the two electrons*), we arrive at the following result for the probability of emission of electrons with total energy \(\varepsilon_1=\dfrac{E}{mc^2}\) and \(\varepsilon_2=\dfrac{E_0-E}{mc^2}\) (\(E_0\) is the decay energy):
\[
w(\varepsilon_1)\,d\varepsilon_1=
\frac{g^4}{2^9\pi^5 R^2}\,|M|^2 F(\varepsilon_1)F(\varepsilon_2)
\left[(\varepsilon_1-1)^{1/2}(\varepsilon_2-1)^{1/2}\right]\times
\]
\[
\times(\varepsilon_1\varepsilon_2-1)(\varepsilon_1-\varepsilon_2)^2\,d\varepsilon_1,
\tag{22}
\]
where \(|M|\) is the nuclear matrix element.
In this expression we take approximate account of the presence of the Coulomb field by means of the factors \(F(\varepsilon)\) customary in the theory of \(\beta\)-decay. For our purposes, in view of the considerable uncertainty in the estimate of the nuclear element, one may take \(F(\varepsilon)\) equal to the square of the electron wave function at the origin and put
\[ F(\varepsilon)=\frac{2\pi\eta}{[1-\exp(-2\pi\eta)]}, \tag{23} \]
where
\[ \eta=\frac{Ze^2}{\hbar v} \tag{24} \]
and \(v\) is the electron velocity.
For sufficiently heavy elements (for which double \(\beta\)-decay alone is possible) one may simply put
\[ F(\varepsilon)=\frac{2\pi Ze^2}{\hbar v}\simeq \frac{2\pi Ze^2}{\hbar c}\simeq \frac{Z}{22}. \tag{25} \]
To obtain the total decay probability it is necessary to integrate (22) over the electron spectrum. Replacing the factors \(F(\varepsilon)\) by a certain mean value, after a somewhat lengthy calculation we obtain for the half-life:
\[ \frac{1}{T_{1/2}}=\frac{1}{\ln 2}\frac{g^4}{2^9\pi^5}\frac{1}{R^2}\,[F(E_{\mathrm{cp}})]^2 f(x), \tag{26} \]
where
\[ f(x)=\frac{x^4}{3}\left(1+\frac{11}{10}x+\frac{1}{5}x^2+\frac{1}{70}x^3\right) \tag{27} \]
and
\[ x=\frac{E}{mc^2}-2 \tag{28} \]
(\(E\) is the kinetic energy of both electrons).
*) The question of the angular correlation was considered (for variant \(T\)) by Primakoff\(^8\). With present-day technique it is hardly accessible to experimental verification.
The values of the function \(f(x)\) are given in Table I.
Table I
Values of the function \(f(x)\)
| \(x=\dfrac{E_0}{mc^2}-2\) | \(f(x)\) | \(x=\dfrac{E_0}{mc^2}-2\) | \(f(x)\) |
|---|---|---|---|
| 0.5 | 0.0337 | 4.0 | 812 |
| 1.0 | 0.771 | 5.0 | 2 770 |
| 1.5 | 5.31 | 6.0 | 7 730 |
| 2.0 | 21.94 | 7.0 | 18 700 |
| 2.5 | 68.0 | 8.0 | 40 800 |
| 3.0 | 175 | 9.0 | 82 000 |
| 3.5 | 396 | 10.0 | 154 000 |
Introducing, instead of the constant \(g_0\), the quantity\(^*\)
\[ B_g=\frac{2\pi^3\hbar^7\ln 2}{g^2m^5c^4}, \tag{29} \]
we finally obtain
\[ T_{1/2}=\frac{2^7}{\pi\ln 2}\,B_g^2\left(\frac{R}{\dfrac{\hbar}{mc}}\right)^2 \frac{mc^2}{\hbar}\frac{1}{|M|^2}\,[F(E_{\mathrm{cp}})]^{-2}\frac{1}{f(x)}. \tag{30} \]
The quantity \(B_g\) is determined from an analysis of the data on \(\beta\)-decay; it is equal to
\[ B_g=2.6\cdot 10^3\ \text{sec}. \tag{31} \]
Substituting the numerical values of the quantities entering formula (30) and taking \(R=1.2\cdot 10^{-13}A^{1/3}\), we obtain:
\[ T_{1/2}=k\,\frac{A^{2/3}}{Z^2}\,\frac{1}{|M|^2}\frac{1}{f(x)}, \tag{32} \]
where
\[ k=1.3\cdot 10^{27}\ \text{sec.}=4.1\cdot 10^{19}\ \text{yr.} \tag{33} \]
\(^*\) The quantity \(B_g\) is usually introduced now into the theory; it determines the value of \(ft\) for \(\beta\)-decay:
\[ ft=B_g\left(\frac12|\!\int 1|^2+\frac12|\!\int \sigma|^2\right)^{-1}. \]
The values of the coefficients are not strictly established. Gerhart\({}^{39}\), from data on the spectrum of \(\mathrm{O}^{14}\), gives for \(B_g\) the value \(3225\pm75\) sec., and for the coefficient at the second matrix element the quantity \(\dfrac12(1.4\pm0.4)\) (instead of \(\dfrac12\)). These corrections have almost no effect on our estimates.
If approximation (32) is insufficient (for example, for the case of Ca\(^{48}\)), then formula (32) must also be multiplied by
\[ (1-e^{-2\pi\gamma})^2 \simeq (1-e^{-Z/22})^2 . \tag{34} \]
If one further makes the replacement (whose error does not exceed \(\sim 10\%\))
\(A^{1/3} \simeq 1.7Z^{1/3}\), then the formula for (32) assumes its final form:
\[ T_{1/2}\simeq 7\cdot 10^{19} Z^{-1/3}(1-e^{-Z/22})^2 \frac{1}{|M|^2 f(x)} \ \text{years}. \tag{35} \]
Unfortunately, this formula contains the nuclear matrix element, whose value cannot be calculated. We shall try to give only a rough estimate for it.
First of all it is natural to expect that \(|M|\) will be largest in those nuclei in which the spatial part of the nucleon wave function (i.e. the quantum number \(j\)) does not change in the transition, so that the protons formed are found in the same orbits as the initial neutrons. This condition is satisfied only for Ca\(^{48}\) (8 neutrons above the closed shell Ca\(^{40}\) are in the state \(f_{7/2}\)). In other nuclei (see Table III) the number of neutrons is already so large that the protons and neutrons are in different shells. In the case of Ca\(^{48}\) one might expect that the quantity \(M\) would be determined by calculating the matrix element of the operator
\[ \sum_{i\ne k}(1+\sigma_i\sigma_k) \]
between states with isotopic spin 4 (Ca\(^{48}\)) and isotopic spin 2 (the probable value for the ground state of Ti\(^{48}\))*). Such a calculation, however, will not give the correct value of \(M\), as is well known from the example of ordinary \(\beta\)-decay, for which calculation by the shell theory gives the correct result only for cases of mirror transitions. The reason for this discrepancy was indicated by O. Bohr\(^{10}\): it consists in the different form of the “core” of the nucleus (formed by closed shells) in the initial and final nuclei. Owing to the presence of a strong interaction between the “outer” nucleons and the “core,” the form of the nucleus depends on the quantum numbers of the “outer” nucleons. Therefore the wave functions of the “core” at the beginning and at the end turn out to be different, which leads to a decrease of the matrix element. An analysis of the experimental data on \(\beta\)-spectra, carried out by Bohr and Mottelson\(^{10}\), shows that the strong coupling of the “core” and the outer nucleons leads to a decrease of the square of the matrix element for ordinary \(\beta\)-decay by a factor of 50–100 (for example, for the decay of Ca\(^{45}\) this
*) In the \(j^8\) configuration there are 3 states with spin \(I=0\) and \(T=2\), which introduces an additional ambiguity into the calculations.
factor is equal to 0.01, for the decay \( \mathrm{Sc}^{49}\)—0.013, for \( \mathrm{Nd}^{141}\)—0.014, etc.).
Thus, we must split the transition matrix element into two factors. One is the matrix element of the operator
\[ \sum_{i\ne k}\sigma_i\sigma_k \]
(the matrix element of unity is zero for transitions in which the isotopic spin changes), calculated taking into account only nucleons in excess of the filled shell. The second factor takes into account the change in the shape of the nucleus. For an estimate one may assume (this is also the most uncertain factor in the theory) that the square of the second factor is, as in ordinary \(\beta\)-decay, of the order of 0.01. Calculation of the first factor gives for it a value of order unity.*) Under these assumptions and for a decay energy of \(4.3\ \mathrm{MeV}\) we obtain:
\[ T_{1/2}(\mathrm{Ca}^{48}) \simeq 10^{15} - 10^{16}\ \text{years}. \]
It is interesting to compare this result with the decay time obtained in the variant of the theory \(\nu \ne \tilde{\nu}\), in which double \(\beta\)-decay occurs with the emission of 4 particles. The probability of such a decay was calculated by Mayer[^31]. The final result also contains an uncertain nuclear matrix element which, however, should be close in magnitude to the matrix element in formula (35).**) Therefore the ratio of the times depends only weakly on the matrix element. This ratio turns out to be equal to:
\[ \frac{T_{1/2}(\nu \ne \tilde{\nu})}{T_{1/2}(\nu \equiv \tilde{\nu})} \simeq \frac{\pi^2}{2}\,5\cdot 7\cdot 9 \left(\frac{\hbar}{mcR}\right)^2 \Phi(x) \simeq 2\cdot 10^7 \Phi(x); \tag{36} \]
where
\[ \frac{1}{\Phi(x)} = \frac{1}{f(x)}x^7 \left( 1+\frac{1}{2}x+\frac{1}{9}x^2+\frac{1}{90}x^3+\frac{1}{1980}x^4 \right). \]
The values of \(f(x)\) are given in Table I, and \(\Phi(x)\) for several values of
\[ x=\frac{E}{mc^2}-2 \]
is given in Table II.
For \(\mathrm{Ca}^{48}\) the ratio of the times, calculated according to different theories, is \(\sim 2\cdot 10^4\).
*) These calculations were carried out by L. Maksimov. They also are not entirely unambiguous, since specifying the spin and isotopic spin of \(\mathrm{Ti}^{48}\) does not yet determine its wave function. The assumption that the ground state is the state of lowest seniority gives for the matrix element the value 0.5.
**) The difference is connected with the difference in neutrino energy. In Mayer’s matrix element one cannot neglect the difference between the energies of the nucleus in the initial and intermediate states. In the estimate we shall replace this difference by an average energy \(\sim 1\ \mathrm{MeV}\).
In conclusion, let us note, finally, that the decay of a \(\mu\)-meson with the emission of two neutrinos,
\(\mu^- = e^- + 2\nu\), may be regarded as an example of double \(\beta\)-decay according to the Majorana scheme. If the neutrino and antineutrino are identical, then, in principle, decay without emission of neutrinos is also possible;
Table II
| \(x=\dfrac{E}{mc^2}-2\) | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|
| \(\Phi(x)\) . . . . . . . . . . | 0.7 | \(10^{-2}\) | \(2\cdot 10^{-3}\) | \(10^{-3}\) | \(5\cdot 10^{-4}\) |
emission of neutrinos; for a free \(\mu\)-meson, in order to satisfy conservation of energy and momentum, the decay must be accompanied by the emission of a quantum
\[ \mu^- = e^- + \gamma . \]
In the field of a nucleus, in the case of a \(\mu\)-meson lying on the \(K\)-orbit of a heavy atom, a decay is possible in which the momentum is taken up by the nucleus, while the electron carries away all the energy (about \(100\) MeV). Such processes have not been observed up to the present time\({}^{36}\). Contemporary theory does not permit one to calculate the probability of such a process in Majorana’s theory without arbitrariness in the choice of the integration limit\({}^{37,38}\). Therefore it is impossible to draw a final conclusion as to whether the absence in experiment of direct \(\mu \to e\) decay is a refutation of Majorana’s theory and proof that \(\nu \ne \bar{\nu}\).
6. DOUBLE \(\beta\)-DECAY. FIELD OF INVESTIGATION
Let us now consider what is known experimentally about double \(\beta\)-decay. An analysis of the table of nuclei shows that in nature there exist 65 stable nuclei of a number of elements that are isobaric with nuclei of other elements differing in atomic number by two units. It is among these isobars that one should search for nuclei capable of double \(\beta\)-decay. The initial nucleus, stable in the usual meaning of the word, with atomic number \(Z\), the intermediate, unstable nucleus (\(\beta^-\)-, \(\beta^+\)-radioactive or capable of electron capture), and the final, again stable nucleus with number \(Z+2\), form isobaric triads distributed comparatively regularly among the elements of the periodic system in the interval from sulfur \((\mathrm{S}^{36}—\mathrm{Cl}^{36}—\mathrm{Ar}^{36})\) to mercury \((\mathrm{Hg}^{204}—\mathrm{Te}^{204}—\mathrm{Pb}^{204})\).
Accurate measurements of nuclear masses throughout the entire periodic system have at present been carried out far from completely. For this reason the question of the location of the ground levels for the selected nuclei of the type under consideration, and consequently also the question of the decay energy, remains open in 27 cases out of 65. Among the remaining 38 nuclei, in 19 cases the energy relations exclude the possibility of double β-decay with emission of electrons, while at the same time allowing, in four cases, double β-decay with emission of positrons (or conversion). Such nuclei are:
\[ \mathrm{Sc}^{78} - \mathrm{Kr}^{78}\ (-3.2 \pm 0.2); \]
\[ \mathrm{Mo}^{96} - \mathrm{Ru}^{96}\ (-3.0 \pm 0.4);\quad \mathrm{Pd}^{106} - \mathrm{Cd}^{106}\ (-3.0 \pm 0.2) \]
\[ \text{and}\quad \mathrm{Te}^{124} - \mathrm{Xe}^{124}\ (-3.0 \pm 0.15). \]
The decay energies in MeV are indicated in parentheses. The kinetic energy of the two emitted positrons will, as is known, be 2 MeV less than these values (in the atomic weights of neutral atoms, \(2mc^2 \sim 1\) MeV must be added for the emission of one positron). The remaining energy is too small to lead to an appreciable probability of decay.
The data relating to the other 20 selected triads, for which electron double β-decay is allowed, are collected in Table III. For completeness, the table also includes data corresponding to the possible double β-decay of the α-active long-lived nucleus \(\mathrm{U}^{238}\).
It should be borne in mind that the accuracy of nuclear-mass measurements is often overestimated by many investigators. Therefore the numbers determining the difference of the ground-level energies of the nuclei, given in the penultimate column of the table, should be regarded as approximate. It is possible that in some cases the true values of the energy differences even lie outside the indicated errors. The position of the ground level for the intermediate unstable nucleus is, as a rule, measured with still lower accuracy, and is sometimes altogether unknown. In compiling the table, use was made mainly of the results obtained in recent years by Nier’s group\(^{11,12,13}\) (in the range of \(A\) values from 110 to 148) and by Duckworth’s group\(^{14,15}\) (values of \(A\) from 70 to 110 and from 148 to 160).
In deciding the question of selecting a particular isobar as an object of study, a significant role may be played by the magnitude of the isotopic abundance of the given nucleus in nature. The values of this quantity for the head elements of the selected triads, according to data of Seaborg’s group\(^{16}\), are given in the last column of the table*).
\[ \rule{3cm}{0.4pt} \]
*) A similar (short) table was recently published by McCarthy\(^{15}\).
Table II
| No. | Triad | \(A\) | \(Z\) | Mass difference in MeV | Isotopic concentration of the initial nucleus, % |
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 |
| 1 | Ca | 48 | 20 | 4,3±0,1 | 0,18 |
| 1 | Sc | 48 | 21 | 4,0±0,1 | 0,18 |
| 1 | Ti | 48 | 22 | 0 | 0,18 |
| 2 | Zn | 70 | 30 | 1,6±0,4 | 0,62 |
| 2 | Ga | 70 | 31 | ∼1,6 | 0,62 |
| 2 | Ge | 70 | 32 | 0 | 0,62 |
| 3 | Ge | 76 | 32 | 2,2±0,4 | 7,67 |
| 3 | As | 76 | 33 | — | 7,67 |
| 3 | Se | 76 | 34 | 0 | 7,67 |
| 4 | Se | 82 | 34 | 3,4±0,6 | 9,19 |
| 4 | Br | 82 | 35 | — | 9,19 |
| 4 | Kr | 82 | 36 | 0 | 9,19 |
| 5 | Kr | 86 | 36 | 1,5±0,6 | 17,37 |
| 5 | Rb | 86 | 37 | ∼1,8 | 17,37 |
| 5 | Sr | 86 | 38 | 0 | 17,37 |
| 6 | Zr | 94 | 40 | 1,1±0,6 | 17,40 |
| 6 | Nb | 94 | 41 | ∼2,2 | 17,40 |
| 6 | Mo | 94 | 42 | 0 | 17,40 |
| 7 | Zr | 96 | 40 | 3,5±0,6 | 2,80 |
| 7 | Nb | 96 | 41 | ∼3,0 | 2,80 |
| 7 | Mo | 96 | 42 | 0 | 2,80 |
| 8 | Pd | 110 | 46 | 0,9±0,2 | 13,50 |
| 8 | Ag | 110 | 47 | 2,8±0,2 | 13,50 |
| 8 | Cd | 110 | 48 | 0 | 13,50 |
Continuation of Table III
| No. | Triad | $A$ | $Z$ | Mass difference in MeV | Isotopic concentration of the initial nucleus in % |
|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 |
| 9 | Cd | 116 | 48 | $2,5 \pm 0,2$ | 7,58 |
| 9 | In | 116 | 49 | $2,8 \pm 0,2$ | 7,58 |
| 9 | Sn | 116 | 50 | 0 | 7,58 |
| 10 | Sn | 122 | 50 | $0,6 \pm 0,2$ | 4,71 |
| 10 | Sb | 122 | 51 | $2,0 \pm 0,1$ | 4,71 |
| 10 | Te | 122 | 52 | 0 | 4,71 |
| 11 | Sn | 124 | 50 | $1,95 \pm 0,13$ | 5,98 |
| 11 | Sb | 124 | 51 | $3,0 \pm 0,2$ | 5,98 |
| 11 | Te | 124 | 52 | 0 | 5,98 |
| 12 | Te | 126 | 52 | $1,55 \pm 0,13$ | 18,71 |
| 12 | J | 126 | 53 | $1,2 \pm 0,2$ | 18,71 |
| 12 | Xe | 126 | 54 | 0 | 18,71 |
| 13 | Te | 128 | 52 | $1,85 \pm 0,13$ | 31,79 |
| 13 | J | 128 | 53 | $1,8 \pm 0,2$ | 31,79 |
| 13 | Xe | 128 | 54 | 0 | 31,79 |
| 14 | Te | 130 | 52 | $3,25 \pm 0,13$ | 34,49 |
| 14 | J | 130 | 53 | $3,0 \pm 0,2$ | 34,49 |
| 14 | Xe | 130 | 54 | 0 | 34,49 |
| 15 | Xe | 136 | 54 | $0,6 \pm 0,9$ | 8,87 |
| 15 | Cs | 136 | 55 | — | 8,87 |
| 15 | Ba | 136 | 56 | 0 | 8,87 |
| 16 | Nd | 148 | 60 | $2,4 \pm 0,8$ | 5,72 |
| 16 | Pm | 148 | 61 | — | 5,72 |
| 16 | Sm | 148 | 62 | 0 | 5,72 |
Continuation of Table III
| No. | Triad | \(A\) | \(Z\) | Mass difference in \(M_{\text{ev}}\) | Isotopic concentration of the initial nucleus in % |
|---|---|---|---|---|---|
| 17 | Nd | 150 | 60 | \(4.1 \pm 0.9\) | 5.60 |
| 17 | Pm | 150 | 61 | — | 5.60 |
| 17 | Sm | 150 | 62 | 0 | 5.60 |
| 18 | Gd | 154 | 62 | \(1.7 \pm 0.6\) | 22.53 |
| 18 | Tb | 154 | 63 | — | 22.53 |
| 18 | Dy | 154 | 64 | 0 | 22.53 |
| 19 | Gd | 160 | 64 | \(3.3 \pm 1.8\) | 21.90 |
| 19 | Tb | 160 | 65 | — | 21.90 |
| 19 | Dy | 160 | 66 | 0 | 21.90 |
| 20 | U | 238 | 92 | \(\sim 1.1\) | 99.28 |
| 20 | Np | 238 | 93 | \(\sim 1.4\) | 99.28 |
| 20 | Pu | 238 | 94 | 0 | 99.28 |
From consideration of the table it is clear that especially attractive objects of investigation are the isotopes \(\mathrm{Ca}^{48}\) and \(\mathrm{Nd}^{150}\), and also, though to a somewhat lesser degree, the isotopes \(\mathrm{Zr}^{96}\), \(\mathrm{Te}^{130}\), \(\mathrm{Se}^{82}\), and \(\mathrm{Gd}^{160}\). In accordance with theoretical estimates, the period of “neutrinoless” decay for the case of \(\mathrm{Ca}^{48}\) or \(\mathrm{Nd}^{150}\) cannot exceed \(\sim 10^{16}\) years. Therefore, establishing the experimental fact that the half-period of double \(\beta\)-decay for the named isotopes exceeds, say, \(10^{18}\) years would be a strong argument in favor of the existence of two types of neutrino. It should be noted, however, that the position of the ground level for the unstable intermediate nucleus of the triad \(\mathrm{Nd}—\mathrm{Pm}—\mathrm{Sm}\) has not been established, while its position in the triad \(\mathrm{Ca}—\mathrm{Sc}—\mathrm{Ti}\) makes possible the occurrence of the decay process in two stages. These circumstances must, of course, complicate the analysis of experimental data that may be obtained in the investigation of the isotopes \(\mathrm{Ca}^{48}\) and \(\mathrm{Nd}^{150}\). An additional factor complicating the investigation in the case of \(\mathrm{Ca}^{48}\) is the low abundance-
ness of this isotope and, consequently, the need for strong preliminary enrichment of the sample under study, which in turn makes it difficult to obtain the necessary quantities of Ca\(^{48}\).
The listed complications practically disappear when the isotopes Zr\(^{96}\) and Te\(^{130}\) are used, but the expected half-lives increase somewhat in connection with the smaller value of the decay energy.
7. DOUBLE β-DECAY. SEARCHES FOR THE PHENOMENON
Eleven works have been devoted to attempts at the experimental detection of double β-decay; they differ greatly in the choice of method, in the care with which they were carried out, and in the reliability of the results obtained.
The main difficulty arising in the experimental investigation of the question is connected with the exceptionally small probability of the corresponding event and, consequently, with the need to make observations over long intervals of time under conditions of a maximally reduced background. Hence the enormous number of photographs that have to be taken if a Wilson chamber is chosen as the recording method. Hence also the need to use stable and long-operating coincidence circuits if the method of scintillation counters is applied. Hence, too, the attempts to make use of geological data in order to increase the duration of the “experiments” to times on the scale of \(10^9\) years.
Searches for double β-decay began in 1949 with a note by Fireman\(^{18}\), in which, it seemed, the fact of double decay of the tin isotope Sn\(^{124}\), with a half-life \(\sim 5\cdot 10^{15}\) years, had been established with sufficient persuasiveness. It is difficult to say what guided the author of the work in choosing Sn\(^{124}\) as the object of investigation. Neither the decay energy, nor the initial isotopic concentration, nor any considerations connected with other physical properties of the Sn\(^{124}\) nucleus make the choice of this isotope particularly fortunate. However, the interest aroused by Fireman’s work predetermined the direction of several subsequent investigations, and in them systematic searches were continued for the supposedly already discovered double β-decay of tin. These works were not crowned with success, and three years later the author himself acknowledged the erroneousness of his first results, attributing them to contamination of the investigated sample by radioactive impurities\(^{19}\).
The outcome of the work carried out cannot, of course, be assessed as purely negative. In the course of the investigations, interesting and varied experimental methods were applied; these same methods were later used in searches for this phenomenon among other isotopes; a minimal estimate of the half-life of double-
of β-decay was obtained in a number of cases with known reliability.
Let us consider the content of the published works. The noted variety of methods and the difficulty of the investigation make it necessary to discuss the results obtained in all the works without exception.
a) Tin 124. Fireman, 1949.¹⁸
A plate of tin enriched with the isotope Sn¹²⁴ to a concentration of 54%, and a standard sample with the natural isotopic concentration, were placed between a pair of end-window gas counters, surrounded by a system of other counters connected in an anticoincidence circuit to reduce the background. The observed excess in the count of coincident pulses in the end-window counters in the case of the enriched sample was attributed to double β-decay with a half-life from \(4\cdot 10^{15}\) to \(9\cdot 10^{15}\) years. As has already been said, these results proved to be erroneous and were refuted by the author himself in a later work.
b) Tin 124. Lawson, 1951.²⁰
A sample containing 0.27 g of tin enriched with the isotope Sn¹²⁴ to a concentration of 83% was placed inside a Wilson chamber. In all, 8794 photographs were taken with the chamber; the obtained photographs were examined twice; 39 photographs on which 2 electron tracks had been found were again carefully analyzed using stereoscopic projection. However, in no case could the observed double tracks be attributed to double β-decay, since they possessed one of the following defects: the tracks did not have a common origin, were of different “ages,” or belonged to one and the same electron that had undergone scattering. In addition, the directions of the tracks were very close to one another, and the total energy of both electrons was about 0.5 MeV. Taking into account the geometry of the experiment and the sensitive time of the chamber, the authors come to the conclusion that the five observed events should have corresponded to a half-life close to \(10^{16}\) years. Since in fact not a single event was recorded, \( \tau_{1/2} > 10^{16} \) years.
c) Tin 124. Kalkstein and Libby, 1952.²¹
In the work two samples were studied—one enriched with the isotope Sn¹²⁴ to 95% (3 g of tin oxide SnO) and a control sample with the natural content of Sn¹²⁴. Both samples were placed in turn between two gas counters with grid walls (the construction of the counters is described in work¹⁶), and under these conditions
PROPERTIES OF THE NEUTRINO AND DOUBLE β-DECAY
a count was made of coincident pulses in both counters. To reduce the background, the apparatus was surrounded by a system of 11 counters connected in an anticoincidence circuit, and by an iron shield 200 mm thick. No differences in the count of the number of coincidences, exceeding the measurement errors, were recorded in the work with the investigated and control samples.
Assuming that differences in the count smaller than half the standard error cannot be detected, the authors conclude that the differences in the count caused by the effect do not exceed this value. The estimate of the half-life obtained from this is \(\tau/2 > 2 \cdot 10^{17}\) years.
d) Tin 124. Pearce and Derby, 1952.^22
The sample investigated, containing 0.2 g of tin enriched in the isotope \(\mathrm{Sn}^{124}\) to a concentration of 95%, was placed between scintillation counters with anthracene crystals. The thickness of the sample was \(0.1\ \mathrm{g/cm^2}\); the background was recorded when the sample was replaced by aluminum foil; the apparatus was shielded by a layer of lead 160 mm thick. The scintillation counters were connected in a coincidence circuit; coincident pulses were summed by means of an electronic circuit and fed to an 18-channel amplitude analyzer.
If, in the case of \(\mathrm{Sn}^{124}\), a “neutrinoless” decay were observed, then the total energy of both emitted electrons would be constant, and on the distribution curve of pulses by amplitudes a maximum should have been found in the region of 2 MeV (the decay energy of \(\mathrm{Sn}^{124}\)). No indications of such a maximum were found on the distribution curve. In Fig. 1, as an illustration, the distribution curve of the number of coincident pulses, obtained by the authors, is shown
Fig. 1. Energy distribution of coincident pulses obtained with a sample of \(\mathrm{Sn}^{124}\), and of background pulses.
with both samples. The experiment lasted 264 hours. A difference in the counting of pulses between the sample under study and the standard exceeding 0.2 pulse per hour would have been easily registered. Hence, taking into account the geometrical efficiency of the apparatus and the mass of the sample, the authors obtain, as a lower estimate of the half-life, the value $\tau/2 \sim 0.5 \cdot 10^{17}$ years.
d) Tin 124. Feherman and Schwarzer, 1952.^19
In this work a controlled Wilson chamber was used, placed in a homogeneous magnetic field. The chamber was filled with a mixture of helium and ethyl alcohol vapor to a pressure of 108 cm Hg. The chamber was controlled by coincident pulses from a pair of thin-walled counters placed inside the chamber on both sides of the sample, which had the form of a plate $(100 \times 250\ \text{mm})$ and was placed at the center of the chamber. Three samples were studied—Sn$^{124}$ enriched to 95% (weight 2.2 g) and two control samples of natural concentration. The sensitive time of the chamber was about 0.1 sec.; since a coincidence pulse occurs on average once every 2.5 minutes, one controlled photograph was equivalent to approximately 1500 photographs taken at random.
In all, 2654 photographs were obtained with the enriched sample and 1662 control photographs—with samples of natural composition.
The photographs, depending on the form of the registered tracks, were classified by the authors into the following groups: mesons, electrons, scattered electrons, showers, and double electron tracks.
Among more than 2500 photographs pertaining to the enriched sample, the authors find three on which there are tracks of an $s$-shaped form. The form of these tracks agrees with the correct direction of motion of particles from one point of the sample to the periphery of the chamber, as a result of which these tracks may be regarded as evidence of double $\beta$-decay. The total energy of the electrons on these tracks was 0.75; 0.78 and 0.33 Mev. The authors cannot, however, exclude the possibility of multiple scattering of an electron in the gas of the chamber after the electron has passed through the foil. Indeed, in addition to the three named cases, two more tracks of $s$-shaped form were registered in the work, with a sign of curvature opposite to that which would correspond to electrons emerging from the sample. If, nevertheless, it is assumed that the three registered tracks should be attributed to cases of double $\beta$-decay, then the half-life proves to lie in the interval from 2 to $5 \cdot 10^{17}$ years.
e) Tin 124, zirconium 96. McCarthy, 1953.^23
The work is distinguished by the care with which it was carried out and by the serious analysis of the results obtained. A scintillation method was used for recording β-particles; the pulses produced were analyzed by amplitude. The sample under investigation (0.15 g of tin enriched in Sn^124 to 95%, or a powder of ZrO₂ containing 0.5 g of metal enriched in the isotopes Zr^96 or Zr^94 to 89.5 or 97.9%, respectively) was placed between two trans-stilbene crystals of scintillation counters connected in a coincidence circuit. The system was surrounded by four “guard” crystals belonging to two other scintillation counters connected in an anticoincidence circuit.
The arrangement of the elements of the counting system is shown in Fig. 2. To protect against soft cosmic radiation, the apparatus was shielded by a 6-millimeter layer of steel and a 100-millimeter layer of lead; to protect against thermal neutrons, shielding made of layers of cadmium and boron oxide was used.
Labels in the figure: plexiglass light guides; counting crystals; “guard” crystals; sample.
Fig. 2. Schematic representation of the experimental setup.
The sample under study and a reference sample with the natural isotopic composition were placed alternately between the counting crystals for intervals of 10–20 hours. The total duration of individual runs was 100–200 hours. Calibration of both counting systems was carried out every 48 hours; the operation of the amplitude analyzer was checked every 20 hours.
The pulses arriving simultaneously from both counters connected in the coincidence circuit, after amplification, were summed
and were analyzed with a thirty-channel amplitude analyzer. Since the possibility was not excluded that both β-particles might emerge on one side of the sample, in addition to the indicated analysis of coincident pulses, series of measurements were also carried out in which the total number of pulses arriving from one counter was counted, independently of coincidences.
The results obtained in experiments with enriched tin and tin of natural isotopic composition are illustrated by the curves in Fig. 3, which show the distribution of coincident pulses by amplitude. As is seen from the figure, within the limits of error
Fig. 3. Distribution of coincident pulses by amplitudes, obtained with samples of \( \mathrm{Sn}^{124} \) and a natural mixture of tin isotopes.
there is no difference between the two curves. When comparing the distribution curves for the total number of pulses in the region from 2.0 to 3.5 MeV, a slight excess in the count is observed in the case of the enriched sample, which the authors attribute to contamination. The estimate of the lower value of the half-life is found to be close to \(1.5 \cdot 10^{17}\) years.
Thus, the result of the series of works devoted to the search for the double β-decay of \( \mathrm{Sn}^{124} \) is the establishment of a lower limit for the half-life in the region of several units times \(10^{17}\) years. This result, obtained by various methods in six works, may be regarded as established with sufficient reliability.
The experimental data relating to zirconium are presented in the form of distribution curves for the total number of pulses and the number of coincident pulses in Fig. 4. As is seen from the distribution curves for the total number of pulses, for the \( \mathrm{Zr}^{96} \) sample in the region 3.5–4.5 MeV a small excess is observed; on
Fig. 4. Distribution of the counting intensity of the total number of pulses (a) and of the number of coincident pulses (b) by amplitudes for samples \(Zr^{96}\) and \(Zr^{94}\).
In the plots:
- solid line — \(Zr^{96}\);
- dashed line — \(Zr^{94}\).
Axes:
- vertical axis in (a): number of pulses per 218 hours;
- vertical axis in (b): number of pulses per 212 hours;
- horizontal axis: analyzer channel numbers;
- lower horizontal scale: \(\beta\)-energy, MeV.
curves corresponding to counting under coincidence conditions, this excess is almost imperceptible. The author of the paper discusses a number of possible causes of this excess in the count and comes to the conclusion that the most probable cause of the effect is double $\beta$-decay of $\mathrm{Zr}^{96}$ with a period $\sim 0.6 \cdot 10^{17}$ years. At the same time, the large statistical errors of the experiment compel the author to formulate the final conclusion with still greater caution: “the experiments performed indicate, but do not prove, that double $\beta$-decay in the case of $\mathrm{Zr}^{96}$ can occur without emission of a neutrino.” Let us also note that, according to the data of Table I and taking into account the inevitable lowering and spread of the pulse amplitudes due to absorption in the sample and oblique tracks in the crystal, the excess in the count should have appeared rather in the region from 2.5 to 3.5 MeV than from 3.5 to 4.5 MeV.
g) Palladium 110. Winter, 1951.²⁴
A sample of metallic palladium of natural isotopic composition, weighing about 250 g, was placed in a Wilson chamber operating without control. Among more than 10,000 photographs, in two cases double electron tracks emerging from one point of the sample were recorded. If these tracks are regarded as evidence of double $\beta$-decay, then for the half-life period of $\mathrm{Pd}^{110}$, according to the author’s estimate, a value $\sim 6 \cdot 10^{17}$ years is obtained. It is quite possible, however, that these tracks are due to the cosmic background.
The assumption that the observed tracks are due to double $\beta$-decay is contradicted by the following circumstance. The energies of the electrons recorded on one of the photographs are, according to the measurements of the author of the article, 0.8 and 1.4 MeV, and on the other photograph 1.4 and 2.1 MeV. Meanwhile, according to recent measurements (see the data of Table I), carried out after the publication of the note under consideration, the total decay energy in this case should not exceed 1.0–1.1 MeV.
h) Fremlin and Walters, 1952.²⁵
In this work a large number of different substances were investigated with respect to double $\beta$-decay. For the purpose of registering $\beta$-particles, specially prepared electron-sensitive plates were used. The experiments were carried out at a depth of about 560 meters below the surface of the earth, and the plates were additionally protected by iron and lead shields. Under these conditions it was possible to reduce the background from cosmic radiation, as well as from the natural radioactivity of rocks, from 200 to 2.5 tracks per day per 1 mm² of 200 $\mu$ emulsion. The substance under investigation was not introduced into the emulsion, but was placed inside the corresponding
corresponding recess in a graphite block, which was brought into direct contact with the plate. Special control experiments showed that the graphite blocks were completely free of any radioactive contamination. The plates were exposed for two months, and throughout this time they were kept in a nitrogen atmosphere at a pressure somewhat greater than atmospheric. A definite level of humidity was maintained with the aid of phosphorus anhydride. Under these conditions the plates retained high sensitivity and no accidental image appeared in them.
In the exposed plates the number of electron tracks was compared in adjacent regions of the plates situated opposite the sample under study and opposite the graphite block. Since the samples under study (except for lead) had a natural isotopic composition, the observed excess in the number of tracks could not be unambiguously ascribed to any definite isotope.
In addition to counting electron tracks, the number of tracks from \(\alpha\)-particles was also counted; this was regarded by the author as evidence of radioactive contamination of the given element.
In three cases (lead, barium, and osmium) a strong \(\alpha\)-background was unquestionably present, and the noticeable excess in the number of electron tracks is therefore regarded as a consequence of the same contamination. Among the remaining 13 samples, only in one case (molybdenum) can the observed excess in the number of tracks (\(1.2 \pm 0.1\) tracks per \(1\ \mathrm{mm}^2\) per day), in the authors’ opinion, be considered significant. If this effect is attributed to double \(\beta\)-decay of \(\mathrm{Mo}^{98}\) or \(\mathrm{Mo}^{100}\) (the decay energy is unknown in both cases), then for the half-life one obtains a value \(\sim 1.5 \cdot 10^{16}\) years.
For all the remaining elements (calcium, chromium, iron, nickel, zinc, germanium, strontium, cadmium, tellurium, tungsten, platinum) the lower estimates of the half-lives obtained by the authors are \(10^{16}—10^{17}\) years.
In considering the results of this work, it should be borne in mind that the method used, attractive in its simplicity and sensitivity, nevertheless gives no proof that the recorded tracks are due to a double \(\beta\)-decay event.
i) Tellurium 130. Inghram and Reynolds, 1950 ^{26}
Searches for double \(\beta\)-decay \(\mathrm{Te}^{130} — \mathrm{Xe}^{130}\) were carried out in this work by geological methods. Bismuth telluride \(\mathrm{Bi}_2\mathrm{Te}_3\), obtained from Swedish ore deposits, was used; the age of these tellurides is known and is \(1.5 \pm 0.5 \cdot 10^9\) years.
There is, however, a certain additional uncertainty in the “xenon age” of the minerals, associated with the possibility of later crystalline alterations of these minerals under the action of surface waters penetrating to depth.
Ore samples (with a \(Bi_2Te_3 \sim 70\%\) content) were extracted from a depth of about 240 meters. In the opinion of the geologists, it is extremely improbable that the crystals underwent subsequent alterations. A portion of the ore, containing 124 g of tellurium, was crushed and heated in vacuum to a temperature sufficient for decomposition of the mineral and vigorous boiling of the molten bismuth and tellurium.
The inert gases (argon and xenon) released during boiling were collected, purified, and analyzed on a mass spectrometer. The amount of xenon obtained was only \(2.6 \cdot 10^7\ \mathrm{cm}^3\) at atmospheric pressure. The results of the isotopic analysis of xenon are given in Table IV.
Table IV
| \(A\) | 124 | 126 | 128 | 129 |
|---|---|---|---|---|
| Normal (atmosphere) | 0.1% | 0.1% | 2.0% | 26.2% |
| From \(Bi_2Te_3\) | \(<0.2\%\) | \(<0.3\%\) | \(<0.8\%\) | 56.3% |
| The same, reduced to \(Xe^{132}\) | \(<0.7\%\) | \(<0.9\%\) | \(<2.4\%\) | 172% |
| \(\dfrac{Xe\ \text{from}\ Bi_2Te_3 - Xe_{\mathrm{atm}}}{Xe_{\mathrm{atm}}}\) | ? | ? | ? | 5.6 |
| \(A\) | 130 | 131 | 132 | 134 | 136 |
|---|---|---|---|---|---|
| Normal (atmosphere) | 4.1% | 21.2% | 26.9% | 10.5% | 8.9% |
| From \(Bi_2Te_3\) | 3.7% | 23.0% | 8.8% | 3.7% | 3.2% |
| The same, reduced to \(Xe^{132}\) | 11.3% | 70% | 26.9% | 11.3% | 9.7% |
| \(\dfrac{Xe\ \text{from}\ Bi_2Te_3 - Xe_{\mathrm{atm}}}{Xe_{\mathrm{atm}}}\) | 1.75 | 2.3 | \(\equiv 0\) | 0.08 | 0.09 |
As is evident from the data given in the table, the isotopic composition of xenon extracted from the ore differs most sharply from the isotopic composition of atmospheric xenon. The authors of the paper believe that \( \mathrm{Xe}^{132} \), which is contained in the ore, can only be of atmospheric origin. By normalizing to this isotope it is easy to verify that the content of all the remaining isotopes, except \( \mathrm{Xe}^{129} \), \( \mathrm{Xe}^{131} \), and \( \mathrm{Xe}^{130} \), proves to be close to normal. The content of the three named isotopes in the ore xenon, however, sharply exceeds the natural content. The excess of \( \mathrm{Xe}^{129} \) and \( \mathrm{Xe}^{131} \) may be attributed to the processes
\[
\mathrm{Te}^{128}(n,\gamma)\mathrm{Xe}^{129}
\]
and
\[
\mathrm{Te}^{130}(n,\gamma)\mathrm{Xe}^{131}.
\]
Neutron fluxes of increased intensity, whose existence must be assumed in order that, over geological time, the corresponding excess amount of \( \mathrm{Xe}^{129} \) and \( \mathrm{Xe}^{131} \) could accumulate, may be due to the presence, near the ore under study, of mineral deposits rich in uranium. Such deposits (tucholite) do indeed exist. Finally, the only cause of the increased content of \( \mathrm{Xe}^{130} \), in the authors’ opinion, can be the double \(\beta\)-decay of \( \mathrm{Te}^{130} \). Accepting this hypothesis, on the basis of the excess of \( \mathrm{Xe}^{130} \) found, the authors arrive at a value of the half-life of approximately \(10^{21}\) years.
The observed value of the half-life agrees, in order of magnitude, with the value calculated for this element in the scheme
\[
\nu \not\equiv \tilde{\nu}^{*}.
\]
It is possible, however, that the process proceeds by way of two successive decays
\[
\mathrm{Te}^{130}\to \mathrm{J}^{130}\to \mathrm{Xe}^{130}.
\]
The decay energy for the first stage (according to the data of Table III) is
\[
0.25\pm0.25\ \mathrm{MeV},
\]
which leads to times comparable with those observed in the experiment. It does not appear possible experimentally to exclude the variant of successive decays.
Let us note that if it nevertheless turns out that
\[
\nu\equiv\tilde{\nu},
\]
then the result of the experiments should be associated with a very small nuclear matrix element for the transition
\[
\mathrm{Te}^{130}-\mathrm{Xe}^{130}.
\]
k) Uranium \(^{238}\). Levin, Djiorso, and Seaborg, 1950. \(^{27}\)
The essence of the work consisted in searches for 90-year \( \mathrm{Pu}^{238} \), which could have been formed from \( \mathrm{U}^{238} \) by double \(\beta\)-decay. For this purpose, from \(14\ \mathrm{kg}\) of very pure uranium oxide \( \mathrm{UO}_3 \), stored for six years, plutonium was extracted. The complicated chemical procedure consisted of 5 cycles of ether extraction with lanthanum fluoride as carrier. The final sample, containing about \(50\ \mu\mathrm{g}\) of carrier, was tested for the presence of \(5.51\ \mathrm{MeV}\) \(\alpha\)-particles from \( \mathrm{Pu}^{238} \). The count was
\[
0.00\pm0.01
\]
counts per minute. From these data it follows that
the half-life of \(U^{238}\) is greater than \(6\cdot 10^{18}\) years. The authors note that, using tons of ore while maintaining the same sensitivity, one can reach values \(t/2\sim 10^{22}\) years.
l) Calcium\({}^{48}\). McCarthy
Indications have appeared in the literature\({}^{28}\) that McCarthy carried out experiments devoted to the search for double \(\beta\)-decay in the case of \(Ca^{48}\). According to reports, McCarthy had at his disposal a Ca sample weighing \(72\ mg\), with a \(Ca^{48}\) concentration of \(89\%\). The number of coincidences observed in two scintillation spectrometers leads to a half-life of \((5\pm2)10^{16}\) years and a maximum \(\beta\)-particle energy of \(3.7\pm0.5\ Mev\). The number of decay events for the indicated half-life should have been about one per hour. The theoretical value for such an energy is \(\sim 10^{16}\) years.
Without a detailed analysis of the arrangement of the apparatus, the experimental curves, and an estimate of the error, it is impossible to judge the reliability of the data presented.
Up to the present time, no article by McCarthy himself has appeared in print.
The possibility is also not excluded that the observed effect was due to successive decays \(Ca—Sc—Ti\). However, this seems unlikely, since there are direct experiments\({}^{29}\) according to which the half-life of the first of the decays is greater than \(2\cdot 10^{16}\) years.
APPENDIX
SOME PROPERTIES OF TRANSFORMATIONS OF THE DIRAC EQUATION
a) Parity (inversion at the origin of coordinates)
Let us consider the Dirac equation for an electron in an electromagnetic field
\[ \left[\left(\frac{\partial}{\partial t}+ie\varphi\right)+\left(\alpha(\nabla-ieA)\right)+im\beta\right]\psi=0. \tag{1} \]
Introducing the usual Pauli matrices:
\[ \begin{aligned} \gamma^i&=-i\beta\alpha^i\quad (i=1,2,3),\\ \gamma^4&=\beta, \end{aligned} \tag{2} \]
we bring equation (1) to the standard form
\[ D\psi\equiv\left\{\gamma^k(\nabla_k-ieA_k)-i\gamma^4\left(\frac{\partial}{\partial t}+ie\varphi\right)+m\right\}\psi=0. \tag{3} \]
Let us carry out the transformation (inversion at the origin of coordinates):
\[ \left. \begin{aligned} x_k &\to -x_k, \qquad t \to t,\\ A_k &\to -A_k, \qquad \varphi \to \varphi \end{aligned} \right\} \tag{4} \]
and find such a transformation of the function \(\psi\) that equation (3) remains unchanged. Put
\[ \psi \to a_s u_s \psi', \tag{5} \]
where \(u_s\) is a matrix and \(a_s\) is a number. It is evident that \(a_s^2 u_s^2=\pm 1\), since a reflection at the origin of coordinates performed twice can only change the sign of the spinor. For definiteness, assume that
\[ \left. \begin{aligned} a_s^2&=\pm 1\\ \text{and}\qquad u_s^2&=1 \quad \text{(the unit matrix).} \end{aligned} \right\} \tag{6} \]
Carrying out in (3) the substitutions (4) and (5), we see that, for invariance of the equation, it is necessary and sufficient that, when the matrix \(u_s\) is transferred to the left of the operator \(D\), the sign before \(\gamma^k\) change, while that before \(\gamma^4\) remain unchanged. For this it is necessary that
\[ \left. \begin{aligned} \gamma^k u_s&=-u_s\gamma^k,\\ \gamma^4 u_s&=\phantom{-}u_s\gamma^4. \end{aligned} \right\} \tag{7} \]
The conditions (6) and (7) are, evidently, satisfied by the matrix \(\gamma^4\). Thus, under reflection at the origin of coordinates, the wave function is transformed according to the law
\[ \psi \to I_s\psi=a_s\gamma^4\psi, \]
where
\[ a_s^2=\pm 1. \]
b) Charge conjugation
Let us now consider the operation of charge conjugation. The invariance (more precisely, covariance) of the equation with respect to charge conjugation means the following. If we change the sign of the external field, then there exists a simultaneous transformation of the wave function under which the form of the equation remains unchanged. Such a wave function evidently describes the behavior of an “antiparticle,” i.e., a particle having the opposite charge, but otherwise identical with the original particle. Let us find
the indicated transformation. To do this, let us rewrite equation (3), making in it the preliminary substitution
\[ A_k \to -A_k,\quad \varphi \to -\varphi: \tag{8} \]
\[ \left\{\gamma^k(\nabla_k+ieA_k)-i\gamma^4\left(\frac{\partial}{\partial t}-ie\varphi\right)+m\right\}\psi'=0; \tag{9} \]
\(\psi'\) is a “charge-conjugate” function describing a particle with charge of the opposite sign.
In addition, let us write the equation for the “conjugate” function \(\bar\psi\), satisfying the equation
\[ D^{+}\psi \equiv \bar\psi\left\{\gamma^k(\nabla_k+ieA_k)-i\gamma^4\left(\frac{\partial}{\partial t}-ie\varphi\right)-m\right\}=0. \tag{10} \]
Equation (10) is obtained by passing in equation (3) to complex-conjugate quantities and transposing. Equations (3) and (10) describe one and the same particle, in contrast to equation (9), which describes its antiparticle. (The function \(\bar\psi\) is written on the left in order to emphasize the matrix character of the multiplication of \(\bar\psi\) and the matrices \(\gamma\). The operations \(\gamma^k\) and \(\dfrac{\partial}{\partial t}\) act on the function in the usual way.)
As is known,
\[ \bar\psi=\psi^*\gamma^4 \tag{11} \]
(\(\psi^*\) is the complex-conjugate function). This is easily verified by substitution.
Using the transposed matrices
\[ \gamma_{\alpha\beta}^{T}=\gamma_{\beta\alpha}=\gamma_{\alpha\beta}^{*} \tag{12} \]
(the last equality follows from the Hermitian character of the matrices \(\gamma\)), we can rewrite (10) in the form
\[ \left\{-\gamma^{kT}(\nabla_k+ieA_k)+i\gamma^{4T}\left(\frac{\partial}{\partial t}-ie\varphi\right)+m\right\}\bar\psi=0. \tag{13} \]
Comparing this equation with (9), we see that the transition from equation (3) to the charge-conjugate equation is very similar to the transition to equation (9). Put
\[ \psi'=C\bar\psi=aC\psi^*. \tag{14} \]
Unlike inversion of the origin of coordinates, the operation of charge inversion contains a transition to the complex-conjugate function.
PROPERTIES OF THE NEUTRINO AND DOUBLE \(\beta\)-DECAY
Thanks to this, conditions (6) are here replaced by the conditions
\[ C^{+}C=1;\quad |a_C|^2=1;\quad u_Cu_C^{+}=1 \quad (C^{+}\equiv C^{T*}). \tag{15} \]
Substituting (14) into (9) and comparing with (13), we find that in order that the equation for \(\bar{\psi}\) coincide with (13)\(^*\), it is necessary that the following commutation conditions of the matrix \(u_s\) with the matrices \(\gamma\) be satisfied:
\[ \begin{gathered} u_C\gamma^4=-\gamma^{4T}u_C,\\ u_C\gamma^k=-\gamma^{kT}u_C. \end{gathered} \tag{16} \]
Let us note that from these conditions one can obtain the relation between the parities of the particle and antiparticle.
Let the wave function of the particle transform under reflection according to the law
\[ I_s\psi=a_s\gamma^4\psi \tag{17} \]
and, correspondingly, the complex-conjugate function according to the law:
\[ I_s\psi^*=a_s^*\gamma^{4*}\psi^*=a_s^*\psi^*\gamma^4. \tag{18} \]
Then the wave function of the charge-conjugate particle will transform as follows:
\[ I_s(C\bar{\psi})=I_s(C\psi^*\gamma^4) =Ca_s^*\gamma^{4*}\psi^*\gamma^4 =Ca_s^*\gamma^{4*}\bar{\psi}, \tag{19} \]
or, using the commutation rules (16), we obtain:
\[ I_s(C\bar{\psi})=-a_s^*\gamma^4(C\bar{\psi}). \tag{20} \]
Comparing with (17), we see that if the particle has parity \(a_s\), then the antiparticle has parity \(-a_s^*\). For particle parity \(\pm 1\), the parity of the antiparticle will be \(\mp 1\); for particle parity \(\pm i\), the parity of the antiparticle will be \(\pm i\), so that \(a_s(-a_s^*)\equiv -1\) for any choice of \(a_s\).
Let us note that, since the wave functions of the particle \(\psi\) and its charge-conjugate function \(\psi'\) satisfy identical equations only in the absence of an electromagnetic field (in a field the operators differ by the sign of the potential), their linear combination (the Majorana-particle function) can describe only a neutral particle.
The conditions (16), which determine the matrix \(u_C\), can be satisfied explicitly only by choosing some representation of the matrix \(\gamma\), so
\(^*\) These are precisely the invariance conditions, since equation (13) is identical with equation (3).
since relations between \(\gamma\) and \(\gamma^T\) are not invariant in character, but depend on whether the given matrix \(\gamma\) is real (then \(\gamma^T=\gamma\)) or imaginary (then \(\gamma^T=-\gamma\)).
We did not use the explicit form of the matrix \(u_C\) in the calculations; nevertheless we shall give it for the two representations used.
(I) Pauli representation, defined by formulas (2). In this case the matrices \(\gamma^1\) and \(\gamma^3\) are imaginary, and we can write (16) in the form:
\[ \begin{aligned} u_C\gamma^\alpha&=-\gamma^\alpha u_C \quad (\alpha=2,4),\\ u_C\gamma^\beta&=\gamma^\beta u_C \quad (\beta=1,3), \end{aligned} \tag{21} \]
whence
\[ u_C=\gamma^2\gamma^4. \tag{22} \]
The charge-conjugation transformation has the form:
\[ \psi=a_C\gamma^2\gamma^4\bar{\psi}=i a_C\beta\alpha_y\psi^* \tag{23} \]
\((a_C=\pm 1\ \text{or}\ \pm i)\).
(II) Majorana representation. In this representation the matrix \(\gamma^4\) is chosen to be imaginary. Then one may put
\[ u_C=-\gamma^4, \tag{24} \]
and the transformation has the form
\[ \psi'=-a_C\gamma^4\bar{\psi}=-a_C\gamma^4\cdot\psi^*\gamma^4=-a_C\gamma^4\gamma^{4T}\psi^*, \tag{25} \]
or finally
\[ \psi'=a_C\psi^*, \tag{26} \]
i.e. the transition to the charge-conjugate function is simply the transition to the complex-conjugate function (this was also the reason for the choice of sign in (24)).
Let us note that transformations of inversion are often written down omitting the factor \(a_C\) (the “intrinsic” parity of the particle).
In conclusion we summarize the properties of the matrix \(C\) (which obviously do not depend on whether the factor \(a_C\) is included in its definition or not):
\[ C^T=-C, \tag{27} \]
\[ C^+C=1\quad (C^+=C^{T*}), \tag{28} \]
\[ C\gamma^\alpha C^{-1}=-\gamma^{\alpha T}, \tag{29} \]
\[ C\gamma^\alpha\gamma^\beta C^{-1}=(\gamma^\alpha\gamma^\beta)^T, \tag{30} \]
\[ C\gamma^\alpha\gamma^\beta\gamma^\gamma C^{-1}=-(\gamma^\alpha\gamma^\beta\gamma^\gamma)^T, \tag{31} \]
\[ C\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\delta C^{-1}=(\gamma^\alpha\gamma^\beta\gamma^\gamma\gamma^\delta)^T. \tag{32} \]
Formulas (30), (31), and (32) are obtained as consequences of formula (29). The last four formulas define the commutation conditions with the operators used in the theory of β-decay. Formula (28) follows from the fact that \(CC^{+}\) commutes with all \(\gamma\).
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